Re: [math-fun] surfaces of revolution & differential equations
Quoting Veit Elser <ve10@cornell.edu <http://gosper.org/webmail/src/compose.php?send_to=ve10%40cornell.edu>>:
Bill,>> The server is back up and yes, the solutions for 4, 5, 7, 8, ...> disks are examples of symmetry breaking (the radii are not equal> in the configurations that maximize the radius sum).
Eh? 7 is as symmetrical as you can get.
Fourteen appears to be the first instance where all the radii are distinct.
In your aluminum puzzle, two disks are almost indistinguishable, but not switchable, due to the precise machining. But two others *are* switchable and apparently indistinguishable. Are you saying they are mathematically distinct?? By how much? Do you have their actual polynomials? --Bill BtW, the mnemonic "1,4" encodes a seed configuration from which the entire solution follows almost immediately.
Since fourteen is the first number where symmetry is completely> broken, that would make it my favorite number.>> My place of work was founded on a principle expressed in fourteen> words.>> Veit>> On Jun 1, 2011, at 5:30 AM, Bill Gosper wrote:>>> Veit>>>> Is your five disk packing of the unit disk maximizing sum(radii)>>>> (http://gosper.org/HTMLFiles/5disks.gif) an example of either of>>>> these asymmetries? How about your semisecret fourteen disk solution,>>>> which has no symmetry at all? (http://milou.msc.cornell.edu/images/>>>> seems to be down.)>>>> --rwg
On Jun 2, 2011, at 5:15 PM, Bill Gosper wrote:
Eh? 7 is as symmetrical as you can get. I miscounted: 3 and 7 are the only solutions with all equal radii (I'm not including 2 because the solution is a continuum).
Fourteen appears to be the first instance where all the radii are distinct.
In your aluminum puzzle, two disks are almost indistinguishable, but not
switchable, due to the precise machining. But two others *are*
switchable and apparently indistinguishable. Are you saying they are
mathematically distinct?? By how much? Do you have their actual polynomials?
--Bill The two disks you refer to have very slightly different radius, about 0.013%. That's about 0.1 mil at the scale of the aluminum puzzle -- beyond what can be machined or discerned by hands and eyes. I never worked out the polynomials.
Veit
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Veit Elser